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I need to solve the following:

$$\iint_S x^2 z ~d\rho,$$

where $S$ is part of the cylinder $x^2 + z^2 = 1$ that is above the $xy$-plane and between the planes $y = 0$ and $y = 2$.

So it looks like I have portion of the cylinder... but again dont know how to setup the integral. I know I have to put the integrand in parametric form first and then I can plug that into the integrand and proceed to integrate.. but the issue is getting the integral limits and the parametric form!

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It should be above the $xz$-plane not the $xy$-plane. See here. – Mhenni Benghorbal Aug 3 '13 at 10:51

The surface can be parametised by $$x=\cos\theta,\,z=\sin\theta,\,y=y,$$ With $-\pi\leq\theta\leq\pi$, $0\leq y\leq 2$.

From that point, substitution into your integral and evaluation should be fairly straightforward.

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i.e. cylindrical coordinates. – Argon Aug 15 '12 at 23:12
That is exactly right. – Daryl Aug 16 '12 at 0:40

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